What if a roller-coaster loop were bigger?

Some roller coasters go all the way around a loop. Riders go upside down!

What if the loop were twice as big? Would the car still make it around?

A roller coaster car rolls down a big first drop, races into a loop and goes over the top upside down. To make it around, it has to be moving fast enough at the very top.

So what happens if engineers make the loop twice as big, but keep the same first hill? In this lab the car starts 40 meters up. A 10-meter loop is no problem. What about a 20-meter loop?

A loop is a test of speed, not just height. At the top, the track and gravity together must supply the centripetal force that keeps the car moving in a circle, and if the car is too slow, gravity alone is more than enough and the car would leave the track.

This lab uses an energy-and-friction coaster model with a 40 m first drop. It asks what happens when the loop radius is doubled from 10 m to 20 m, and how tall the first drop must then be.

The car rolls down a tall hill and goes very fast.

It zooms around the loop and over the top. Easy!

In the model, the car starts at rest 40 meters up, on a track with a little friction. At the bottom of the drop it is moving at about 98 kilometers per hour.

The loop has a radius of 10 meters, so its top is 20 meters high. The car needs at least 9.9 meters per second at the top to stay on the track. It arrives with 16.8 meters per second, plenty to spare, and it makes the loop.

From 40 m with a friction coefficient of 0.02 (lumping rolling friction and drag), the car reaches about 98 km/h at the bottom. At the top of a 10 m loop, 20 m up, it needs √(g·r) = 9.9 m/s and arrives at 16.8 m/s. The push of the track on riders there is 1.9 g.

That critical speed, the square root of g times r, is the same rule TeachEngineering's roller coaster lesson gives for the minimum speed at the top of a loop.

Now Olly makes the loop twice as big. The top of the loop is as high as the start!

The car can't get to the top. It rolls back down.

Now make the loop's radius 20 meters. Its top is now 40 meters high, exactly as high as the start. The car can't even climb that high, because friction has taken some of its energy, so it never reaches the top and rolls back.

Even without friction it wouldn't work, because the car needs speed at the top, not just height: 14.0 meters per second for this bigger loop. In the model, the first hill must start at least 58 meters up for the 20-meter loop, twice the 29 meters needed for the 10-meter loop.

With r = 20 m the loop top is at 40 m, level with the start, and the needed speed rises to √(9.81 × 20) = 14.0 m/s. With friction the car never reaches the top; even with no friction it would arrive with essentially zero speed.

Scanning start heights in 1 m steps, the model's minimum is 29 m for r = 10 m and 58 m for r = 20 m with μ = 0.02, and 26 m and 51 m with no friction, close to the ideal 2.5 r. At 60 m the car makes the 20 m loop with 15.5 m/s at the top.

At the top of a loop, the car must still be going fast.

A bigger loop is higher, and it needs more speed at the top. So the hill must be taller.

At the top of a loop, the car is upside down and moving in a circle. If it is fast enough, the track keeps pushing on it from above, holding it on its circular path. Too slow, and it would start to fall away from the track. A bigger loop needs a bigger speed at the top.

The car gets that speed from its height. It has to climb to the top of the loop, twice the radius, and still have speed left over. Put those together and, without friction, the start must be at least two and a half times the loop's radius. Double the loop, double the hill.

At the loop top the condition v² ≥ g·r means the track's push is zero or positive. Energy conservation from rest at height H gives v² = 2g(H − 2r), so H ≥ 2.5 r: the minimum start height scales linearly with the radius, while the critical speed scales with √r (9.9 to 14.0 m/s when r doubles).

The model's limits: a point car on a 2D track, a perfectly circular loop, friction and air drag lumped into one coefficient, and no train length. A circular loop also gives very large pushes at its entry (the model reports up to 8.6 g in the 10 m case), which is why real loops are not circles.

Real roller coaster loops are not perfect circles.

They are shaped like a teardrop: wide at the bottom and tight at the top. Riders have wheels under the track too, to keep them safe.

Most real roller coaster loops are not circles. They have a teardrop shape called a clothoid. The curve is gentle at the bottom, where the car is fastest, and tight at the top, where it is slowest. That makes the ride smoother.

TeachEngineering's roller coaster lesson explains that the modern looping coaster arrived in 1976 with Revolution at Six Flags Magic Mountain, which used a clothoid loop. Most coasters also have wheels on both sides of the track, so the cars can't fall off.

In a clothoid, the radius of curvature is largest at the bottom, cutting the push on riders where the speed is highest, and smallest at the top, where the car is slowest, so less speed is needed there. TeachEngineering's Duke University lesson notes that early circular loops caused rider injuries, that looping coasters were abandoned in 1901, and that they returned in 1976 with Revolution at Six Flags Magic Mountain, the first modern looping coaster with a clothoid shape.

It also notes that most coasters carry wheels on both sides of the track. The model checks contact at the top as if those wheels were absent, to show the physics.